The Market Prices Disasters as 1‑in‑8 (Even If History Says 1‑in‑60)
Meyerheim: "Rare Disasters, Tail Aversion, and Asset Pricing Puzzles" CRC Discussion Paper No. 549
A new working paper by Gerrit Meyerheim (LMU Munich) offers a simple synthesis for two classic macro‑finance problems: equities earn a large premium, yet real risk-free rates are low. The punchline is that you do not need extreme risk aversion. Instead, what matters is how strongly prices react to the left tail.
Key numbers
Long‑run real equity return: ≈ 7.1%/year with ≈ 22% volatility (Sharpe ratio ≈ 0.26).
Long‑run real risk‑free rate: ≈ 1.3%/year.
In the paper’s baseline calibration, the risk‑neutral probability of a major disaster is ~13%/year, about 8× the physical probability (≈ 1.7%).
Why this is a puzzle
In the textbook consumption‑based model, consisting of constant relative risk aversion (CRRA) utility and roughly lognormal consumption growth, aggregate consumption is not volatile enough to justify the observed Sharpe ratio unless risk aversion is implausibly high. However, pushing risk aversion up typically drives the model’s risk‑free rate up as well, unless you assume unrealistic levels of impatience. That is the equity premium puzzle plus the risk‑free rate puzzle in a nutshell.
The solution: keep CRRA, change the one‑period pricing weights
1) Rare disasters in consumption. Most years look “normal”, but with small probability consumption experiences a large negative jump.
2) Tail aversion via an entropic tilt. Period utility stays CRRA with standard risk aversion, but the one‑period pricing kernel overweighs bad consumption‑growth states by an exponential tilt. A robust‑control recursion delivers the same object.
The key subtlety: the entropic tilt raises the pricing weight on bad states without changing within‑period curvature, it only introduces an exponential change of measure. That is, period utility remains CRRA as in the textbook model.
3) Valuation compression in disasters. When consumption drops, equity returns drop by more, capturing leverage/default and discount‑rate spikes. This is what makes a consumption disaster feel like an equity crash.
What the baseline calibration implies
Using disaster evidence (disaster probability ≈ 1.7%/year; consumption drop ≈ 29%) and matching long‑run return moments (net risk‑free ≈ 1.3%, net equity return ≈ 7.1%, equity volatility ≈ 22%), the paper backs out:
tail aversion is roughly the same order as standard risk aversion,
an implied equity disaster loss ≈ 44% on impact,
a variance risk premium ≈ 0.03 (close to option‑based estimates of around 0.02).
The key result is the wedge between informational and pricing significance:
The one‑period Kullback–Leibler divergence of the tilt is very small; it could take ≈ 40 years of annual data to cleanly tell the model apart from standard CRRA.
Yet the same parameters imply a risk‑neutral disaster probability ≈ 13%. That is not an actual belief; it is the probability embedded in prices.
Options, almost for free
Under the risk‑neutral measure, equity returns become a simple two‑state mixture (normal vs. disaster). That means one‑period index puts can be priced as a convex combination of two Black–Scholes formulae, yielding a naturally downward‑sloping implied‑volatility “smirk” without assuming ad‑hoc stochastic volatility.
Takeaway
If standard CRRA is decent at pricing “ordinary risk” but struggles with “catastrophe insurance,” this paper’s message is: keep CRRA, add rare disasters with realistic valuation compression when disasters hit, and let the one‑period pricing kernel lean slightly harder into the left tail. A small, disciplined tail tilt goes a long way toward resolving the big puzzles and connecting them to option‑market tail‑risk facts.
Link (pdf): Rare Disasters, Tail Aversion, and Asset Pricing Puzzles
